Interestingly, we now mapped the length scales competition between the 1D lattice and the quasiperiodic potential in Eq. (REF ) to a competition between the 2D lattice and the magnetic length scale in Eqs. (REF ) and (REF ). In the absence of the magnetic field, \(\tau \rightarrow \infty \) , the Hamiltonian (REF ) commutes with the group of lattice translations, whereas when the magnetic field breaks this
symmetry, the notion of the magnetic translation group was introduced [1]}. It is generated by
the operators \(T_{\hat{m}}\)  and \(T_{\hat{n}}\) , where \(T_{\hat{m}} c_{n,m} T_{\hat{m}}^{-1} = c_{n,m+1}\)  and \(T_{\hat{n}} c_{n,m}T_{\hat{n}}^{-1} = e^{-i2\pi m/\tau } c_{n+1,m}\) . For an irrational \(\tau \) , these operators commute with the Hamiltonian but not with each other. Nevertheless, similar to the approximant discussion above, for a rational flux, \(\tau =q/p\) , the operator \(T_{q\hat{n}} = (T_{\hat{n}})^q\)  commutes with \(T_{\hat{m}}\) . We can therefore
simultaneously diagonalize \(H_{\rm 2D}\) , \(T_{q\hat{n}}\) , and \(T_{\hat{m}}\) . Here, too, the enlarged real space magnetic unit cell implies folding of the corresponding quasimometa (now onto a 2D torus) and the formation of \(q\)  bands in the spectrum, thus generating Hofstadter butterfly spectra, see Fig. REF (b) [2]}.
